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Transport Theories for Heavy Ion Collisions in the 1 AGeV

Regime

E.E. Kolomeitsev1, C. Hartnack2, H.W. Barz3, M. Bleicher4, E.

Bratkovskaya4, W. Cassing5, L.W. Chen6,7, P. Danielewicz8, C. Fuchs9,

T. Gaitanos10, C.M. Ko6, A. Larionov5†, M. Reiter4, Gy. Wolf11, J.

Aichelin2‡1 The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen, Denmark,School of Physics and Astronomy, University of Minnesota, Minneapolis, MN-55455, USA2 SUBATECH, University of Nantes - IN2P3/CNRS - EMN, F-44072 Nantes, France3 Forschungszentrum Rossendorf, D-01314 Dresden, Germany4 Institut fur Theoretische Physik der Wolfgang Goethe Universitat, D-60054 Frankfurt,Germany5 Institut fur Theoretische Physik der Universitat Giessen, D-35392 Giessen, Germany6 Cyclotron Institute and Physics Department, Texas A&M University, College Station,TX-77843, USA7 Department of Physics, Shanghai Jiao Tong University, Shanghai 20030, China8 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,MI-48824, USA9 Institut fur Theoretische Physik Universitat Tubingen, D-72076 Tubingen, Germany10 Laboratori Nazionali del Sud INFN, I-95123 Catania, Italy11 KFKI Budapest, POB 49, H-1525 Budapest, Hungary† on leave from I.V. Kurtchatov Institute, 123182 Moscow,Russia

Abstract.

We compare multiplicities as well as rapidity and transverse momentum distributions ofprotons, pions and kaons calculated within presently available transport approaches for heavyion collisions around 1 AGeV. For this purpose, three reactions have been selected: Au+Au at1 and 1.48 AGeV and Ni+Ni at 1.93 AGeV.

1. Introduction

Heavy ion collisions in the energy range from 50 AMeV to several AGeV are quite complex. Thesingle particle spectra are not isotropic and the anisotropy depends significantly on the centralityof the reaction. With increasing energy one observes the rise and fall of multifragmentation,where multifragmentation means that in each single collision a multitude of intermediate massfragments is produced. While above Ekin = 289(1583)MeV , π’s(K+’s) can be produced inelementary nucleon-nucleon collisions, experiments have shown that in collisions between nucleiFermi motion and multiparticle effects allow for meson production at energies well below thethreshold in an elementary reaction. Finally, nucleons can be excited to resonances which -according to observations in γA reactions - change their properties in a nuclear environment.

This complexity of the reactions makes it very difficult to link the experimental results ina unique way with the specific underlying physical processes. The challenge to interpret theexperimental results was first met by Yariv and Frankel [1] and Cugnon [2] who took advantageof the increasing computer power at that time and developed the so-called ”cascade approach”in which nucleons behave similarly to classical billiard balls and scatter with the free elementarycross sections. This computational approach allowed for completely new insights into the reactionmechanism and allowed for the first time to interpret the nonequilibrium features of the observedspectra. Disregarding binding energy and mean field effects, these approaches have been limited tosmall systems at higher energies. The extension to lower energies became possible half a decade laterwith the numerical realization of the Boltzmann-Uehling- Uhlenbeck (BUU) [3] or Vlasov-Uehling-

‡ invited speaker

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 2

Uhlenbeck (VUU) [4] approaches which supplemented cascade approaches by the introduction ofan attractive mean field and of Pauli blocking, which suppresses collisions if the phase space isalready occupied by other nucleons. This made it possible to describe heavy ion reactions downto energies of 20 AMeV. Over the years these approaches became more and more refined as aresult of constructive competitions among different groups and many physical questions have beenaddressed and answered. The pre-equilibrium proton emission has been quantitatively reproduced.It is nowadays established that subthreshold mesons like η, K or ρ are created in elementarycollisions, where at least one of the collision partners had gained additional energy in previouscollisions. It is further established that resonances are the primary source for the produced pions.

In the meantime, the simulation programs have matured to the level which allows to goeven beyond the original goal of describing the elementary reaction features: They are nowadayscommonly used as a tool to find out whether the experimental results allow for a determination ofphysical quantities for which no solid theoretical predictions are available. These quantities includethe energy which is needed to compress nuclear matter as well as the properties of resonances andmesons in a hadronic environment.

For the following three reasons it seems to be useful to compare the simulation programs whichhave been developed over the years:

1) The simulation programs are rather complex - consisting of several thousand program lines- and use quite different numerical techniques. Thus, one has to assess whether the differentnumerical procedures lead to the same results.

2) All programs rely on inputs. Those inputs include all the needed elementary cross sections.Among those there are cross sections which are not known experimentally (like those including abaryonic resonance in the entrance channel), and different theoretical implementations have beenintroduced. It is evident that different elementary cross sections can yield different results.

3) In all programs resonances are produced. Their properties in the hadronic environment aswell as their time evolution are little known. The programs use different parameterizations for thisand it is not a priori evident how those differences influence the final result.

Thus the detailed comparison presented in the present paper may serve as the first steptowards a critical assessment of the predictive power of the simulation programs and to identifythe features which have to be improved. We would like to mention that some elements towardsa future development are already available. Very detailed calculations of the spectral functionof kaons in infinite matter have been published [5] and first steps towards describing the timeevolution of resonances in matter have been made [8, 6, 7]. These advances, based on the gradientexpansion of the Kadanoff-Baym equation, are difficult to formulate for test particles, and furtherapproximations have been shown to be necessary for actual applications [8]. Apart from theseapproaches there are a couple of studies providing information on the off-shell transition rates[9, 10].

2. The simulation programs

In this comparison nearly all the presently available simulation programs have taken part: Thesimulation program from the Budapest/Rossendorf group [11], the HSD program developed inGiessen by Bratkovskaya and Cassing [12, 13], the simulation program developed by Bratkovskaya,Effenberger, Larionov and Mosel in Giessen [14], the code developed by Danielewicz in Michigan[15], the RVUU approaches of the Texas A&M group [16] and the Munich/Catania/Tubingen group[17]. Besides these true BUU or VUU models, there are several models which are based on theQuantum Molecular Dynamics approach. These n-body approaches allow for the description offragment production but as far as the observables discussed here they should give the same results.Therefore, we just refer to ref. [18] for the general features and do not discuss the differenceshere. These simulation models include the IQMD [19] and URQMD [20], both developed in acollaboration between Nantes and Frankfurt, as well as the QMD of the Tubingen group (byFuchs)[21]. These programs contain a different number of baryonic resonances as can be inferredfrom table 1. The URQMD code has been developed for higher (SPS to RHIC) energies andincludes neither a binding energy for nucleons nor a kaon-nucleon potential but all known mesonicand baryonic resonances. Also its production mechanism forK+ is different from other approaches.Therefore it is only included to illuminate the differences.

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 3

Program τ ResonancesBarz τiso ∆′s,N’s with M < 2 GeV

RVUU (Chen) τiso ∆(1232)Danielewicz τiso ∆(1232), N∗(1440)QMD (Fuchs) τiso ∆(1232), N∗(1440)

Gaitanos τiso ∆(1232), N∗(1440)HSD (Cassing) τiso ∆(1232), N∗(1440), N(1535)

IQMD (Hartnack) τwig ∆(1232)Larionov 1/120 MeV ∆′s,N ′s with M < 2 GeV

Table 1. The width of the ∆ resonance and the resonances employed in the programs.

3. The Procedure

The simulations programs, which are compared in this article, have been frequently compared withexperimental data and usually gave quite satisfying agreement. It is, however, difficult to baseprogram comparisons on those calculations because rarely the same data have been compared withdifferent simulation programs and in the rare cases one has done so, it is not evident that the sameexperimental filter or the same centrality selection has been employed. To assess the predictivepower of these programs and to see whether differences in the predictions are sufficiently largeto be experimentally relevant, it is much better to compare the results of the programs directlywithout any cuts. In addition, all programs employ the impact parameter as an input variable andtherefore it is most convenient to compare the results at a given impact parameter. This excludesof course any comparison with experiment. In this publication we concentrate on 3 reactions: Au+ Au at 1 and 1.48 AGeV and Ni+Ni at 1.93 AGeV, all at an impact parameter of b = 1 fm as wellas the pion yield as a function of the impact parameter. This choice was guided by the availableexperimental results.

4. Protons

The final proton rapidity distribution measures the amount of stopping and reflects how muchenergy becomes available for particle production. The recent analysis of the FOPI collaboration[22] shows that, in the center of mass system, the variance of the transverse rapidity distributionis always smaller than twice the variance of the longitudinal rapidity distribution even in the mostcentral collisions. If the system were in thermal equilibrium one would expect that the ratio R ofthe variances equals one. Experimentally the largest degree of thermalisation (R ≈ 0.85) is observedfor central Au + Au collisions around 400 AMeV. Below that energy Pauli blocking is still tooactive, while above the NN cross section becomes more and more forward peaked and therefore theenergy transfer into the perpendicular directions is lowered. In addition, more energy has to betransfered in order to come to equilibrium. The rapidity and transverse momentum distributionsfrom simulations, figs.1 and 2, reflect this approach to equilibrium. For the heavy systems thefinal rapidity distributions is peaked at or very close to midrapidity and the simulations differonly little. Initially the protons are located around ycm = 0.68, 0.8, 0.89 for the three reactions,respectively. The resulting rapidity shift for the large systems is thus large. A slightly smallershift is observed for the light system and there also differences develop between the predictions ofthe different programs. This kind of differences should be measurable and can help to validate theassumptions made. In transverse direction and on logarithmic scale, we find a general agreementamong the programs for Au+Au collisions. However, in the lighter system pronounced differencesdevelop. They seem to be correlated with the rapidity distribution and reflect a different degreeof stopping in the different approaches. The differences are most pronounced at small transversemomenta. To fill this part of phase space many collisions are necessary. Therefore its form isa measure for the number of collisions during the interaction. Table 2 summarizes the averagesquared momenta in the center of mass system at the end of the reaction, along the beam (z)direction and in the direction (y) perpendicular to the reaction plane.

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 4

0.0 0.5 1.0 1.50

50

100

150

200

p

Au+Au 0.96 AGeV

d

N/d

y

y-yc.m.

0.0 0.5 1.0 1.50

40

80

120

160 Au+Au 1.48 AGeV

y-yc.m.

0.0 0.5 1.0 1.50

10

20

30

40

Barz/Wolf

Cassing

Hartnack

Larionov

Chen

Caitanos

Danielewicz

Ni+Ni 1.93 AGeV

y-yc.m.

Figure 1. Final proton rapidity distribution at b = 1fm in the different approaches

0.0 0.5 1.0 1.5 2.00.1

1

10

100

1000Au+Au 0.96 AGeV

dN

/pTd

pT [1

/Ge

V2]

pT [GeV/c]

p

0.0 0.5 1.0 1.51

10

100

1000

Au+Au 1.48 AGeV

pT [GeV/c]

0.0 0.5 1.0 1.50.1

1

10

100

1000

Barz/Wolf

Cassing

Hartnack

Larionov

Chen

Gaitanos

Danielewicz

Ni+Ni 1.93 AGeV

pT [GeV/c]

Figure 2. Final proton transverse momentum distribution at b = 1fm in the different approaches

1 AGeV Au+Au 1.48 AGeV Au+Au 1.93 AGeV Ni+Ni

Program 〈p2z〉 〈p2y〉〈p2

y〉〈p2

z〉〈p2z〉 〈p2y〉

〈p2y〉

〈p2z〉

〈p2z〉 〈p2y〉〈p2

y〉〈p2

z〉(GeV 2/c2) (GeV 2/c2) (GeV 2/c2)

Barz/Wolf 0.157 0.109 0.69 0.230 0.146 0.64 0.351 0.147 0.42RVUU (Chen) 0.165 0.121 0.73 0.256 0.172 0.67 0.416 0.172 0.42Danielewicz 0.162 0.102 0.63 0.257 0.126 0.49 0.407 0.104 0.26QMD (Fuchs) 0.158 0.111 0.70 0.262 0.138 0.53 0.413 0.107 0.26

Gaitanos 0.131 0.116 0.89 0.240 0.142 0.60 0.300 0.104 0.35HSD (Cassing) 0.138 0.120 0.87 0.219 0.157 0.72 0.361 0.148 0.41

IQMD (Hartnack) 0.143 0.121 0.85 0.226 0.161 0.72 0.378 0.158 0.42Larionov 0.163 0.124 0.76 0.252 0.159 0.63 0.405 0.136 0.34

Table 2. Average out-of-plane and longitudinal squared momenta for protons in the differentreactions.

5. Pions

The direct pion production cross section in nuclear interactions is small. Therefore all pions arecreated in simulations via resonances. These resonances are produced with the free inelastic NNcollisions - which is well known - and in the region of interest comparable to the elastic cross section.In our energy region these are mostly ∆ resonances which disintegrate into a π and a nucleon. Theproblem is how to treat this resonance in matter as is required for the simulation programs. The∆ resonance, as noted in the particle data booklet, has a width Γ of about 120 MeV and hence

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 5

1.1 1.2 1.3 1.4 1.5 1.610

-1

2

5

100

2

5

101

2

5

102

2

5

Figure 3. The different ∆ lifetimes which have been proposed.

a lifetime of τ = 1Γ = 1.7fm/c. Therefore it disintegrates inside the nuclear medium. The width

has been determined by analyzing the phase shift in πN scattering as a function of√s. Around

δ(√s) = π/2 the dependence on the energy may be approximated with δ(

√s) = tan−1( Γ

2(m∆−√s)).

The lifetime as inverse of a constant width Γ is justified if a) the width were small as compared tothe mass difference between the resonance and its decay products (Γ << m∆− (mN +mπ)) and ifb) the energy spread of the particle wave packets were large compared to Γ. This is, however, notthe case: the nucleons are represented by delta functions in energy. Therefore one has to addressthe question which lifetime should be assigned to the object which is created in an inelastic nucleon-nucleon collision with a sharp

√s and which yields two nucleons and a pion in the exit channel.

There are two approaches in the literature. Ericson and Weise [23], in particular, have analyzedthe πN∆ system in a relativistic isobar model and found for the scattering amplitude

f∆33 =

γk2

m2∆ − s− iγ|k|3 , (1)

a form which resembles the usual relativistic Breit-Wigner form and yields an energy dependent ∆decay width of

Γ∆(|k|) = 2γ|k|3 =2f2

∆|k|3M∆

12πm2π

√s. (2)

With f2∆ = 4.02 this form reproduces very well the π N data. Γ∆(|k|) decreases with decreasing√

s and therefore a lifetime defined as

τiso ∝ 1

Γ∆(|k|)(3)

increases with decreasing√s.

On the other hand, the lifetime of a resonant state has been already analyzed by Wigner [24].In recent time this considerations have been generalized by Danielewicz and Pratt [25]. They foundthat quantum mechanically the reaction slows down with a time delay between the outgoing andthe incoming wave of

τwig = 2dδ(E)

dE. (4)

Hence maximal time delay occurs at the center of the resonance. Fig. 3 show the ∆ lifetimes in thedifferent approaches as a function of the ∆ mass. Randrup, Kitazoe and Huber are synonymous forthree rather similar approaches which calculate the width using the p-wave scattering amplitudeas described above. The description used in the standard versions of the different programs areshown in Table 1.

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 6

The influence of the choice of τ on the final number of pions is by far not negligible. Almostindependent of energy and size of the system τiso may yield up to 60% more pions than τwig. If the∆ has a short lifetime, several generations of π and ∆ are produced. In an expanding system theaverage available center of mass energy in the NN system decreases and the cross section for theproduction of new ∆’s becomes smaller whereas for the backward reaction it is little influenced.

Fig. 4 shows the number of produced pions as a function of the impact parameter for thethree reactions. We see the expected significant variations of the total yield. In order to put the

0

5

10

15

20

π+

Au+Au 0.96 AGeV

Multip

licity

0

10

20

30

40

50 Au+Au 1.48 AGeV

0

4

8

12

16

Ni+Ni 1.93 AGeV

0 4 8 12 160

5

10

15

20

25

π--

b [fm]0 4 8 12 16

0

10

20

30

40

b [fm]0 2 4 6 8 10

0

4

8

12

16 Barz/Wolf

Cassing

Hartnack

Larionov

Fuchs

Danielewicz

b [fm]

Figure 4. Final π+ and π− multiplicity as functions of the impact parameter in the standardversions of the different approaches characterized by different descriptions of the ∆ lifetime (seetable 1).

results on an equal footing we have decided that all other calculations presented here use a constantlifetime of τ = 1/Γ,Γ = 120 MeV . The rapidity and transverse momentum distributions, obtainedwith this constant lifetime, are displayed in figs. 5 and 6, respectively. We see, first of all, thatthe differences between the calculations are reduced but not eliminated. The calculation of Reiter,based on URQMD, has to be interpreted with care because in this calculation neither potentialsare present nor - being an approach for high energies - special care has been taken to model theinelastic cross section close to threshold. In the program of Chen only the isospin averaged pionyield is available. For that program we have therefore plotted here for π− and π+ one third ofthe total π yield. Aside from Reiter’s results, the rapidity distributions are rather similar for themajority of the programs. There are, however, some exceptions which show a narrower distribution.The experimental ratio R = π−/π+ at 1 AGeV is close to the value R=1.95 [26] which one expectsfrom the Clebsch Gordon coefficients if (in this neutron/proton asymmetric system) the π’s areexclusively produced via ∆ resonances. Most of the simulation programs reproduce this value. Theslopes of the transverse momentum spectra at midrapidity are very similar as well. The differencesconcern exclusively π’s with very small pT values. This part of the spectra is sensitive to ∆’swith a very small mass. We would like to stress that some of the programs include higher (N∗ or∆∗) resonances whereas others are limited to the ∆ resonance. This may be of importance at thehigher beam energies. Generally the simulation programs seem to overpredict the pion yield. Acompilation of the available data in ref. [26] shows an average total pion yield per participant of

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 7

0

5

10

15

20

π+

Au+Au 0.96 AGeV

d

N/d

y

0

5

10

15

20 Au+Au 1.48 AGeV

0

2

4

6

8

Barz/Wolf

Cassing

Reiter

Hartnack

Ni+Ni 1.93 AGeV

-2 -1 0 1 20

5

10

15

π--

y-yc.m.

-2 -1 0 1 20

10

20

30

y-yc.m.

-2 -1 0 1 20

2

4

6

8

Larionov

Chen

Fuchs

Gaitanos

Danielewicz

y-yc.m.

Figure 5. Final π+ and π− rapidity distributions at b = 1fm in the different approaches withan enforced ∆ lifetime of 1/120 MeV.

0.0 0.2 0.4 0.6 0.80.1

1

10

100

1000

Larionov

Chen

Fuchs

Gaitanos

Danielewicz

Au+Au 0.96 AGeV

π+

dN

/pTdp

T [1/G

eV

2]

pT [GeV/c]

0.0 0.2 0.4 0.6 0.80.1

1

10

100

1000

Barz/Wolf

Cassing

Reiter

Hartnack

Au+Au 1.48 AGeV

pT [GeV/c]

0.0 0.2 0.4 0.6 0.80.1

1

10

100

1000 Ni+Ni 1.93 MeV

pT [GeV/c]

Figure 6. Final π+ transverse momentum distribution at b= 1 fm and |ycm < 0.5| in thedifferent approaches with an enforced ∆ lifetime of 1/120 MeV.

π++π0+π− = 1.5(π++π−) = 0.075 and therefore about 30 pions in the most central collisions ofAu+Au at 1 AGeV. Please note that the slope of the pion transverse momentum spectra is ratherdifferent from that of the protons. This is due the convolution of the ∆ transverse momentumdistribution with its the decay kinematics.

6. K+ Mesons

Fortunately, the uncertainty about the optimal description of the ∆ lifetime influences neither themultiplicity nor the form of the K+ spectra in a significant way because low mass ∆’s have asmall cross section to produce a K+. Due to the high threshold in the mesonic channel the K+

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 8

are mostly produced in baryon-baryon collisions. To make comparisons more straightforward weenforced for the comparison the same cross sections [27] for the N∆ → K+ and the ∆∆ → K+

channels in the different codes. These cross sections are dominant at Ekin ≥ 1.5AGeV . Therefore,and due to the constant ∆ lifetime, the results presented here may differ from already publishedresults. For the NN → K+ cross section we kept the original cross sections of the programs.For the channel pp → K+ new data at low energies are available [28]. In addition there is theproblem of how to extrapolate this pp cross section to the nn, np → K+ channels. Usually thisextrapolation is done by assuming that the K+ is accompanied by a Λ and that only the isospincoefficients change. In the case of the K+ this is, however, not so easy because this extrapolationdepends on whether a kaon or a pion is exchanged. For the case of a pion exchange we findσ(np → K+) = 5/2 σ(pp → K+) and consequently σ(NN → K+) = 3/2 σ(pp → K+); in the caseof a kaon we find σ(pp → K+) = 2σ(np → K+) and therefore σ(NN → K+) = 1/2 σ(pp → K+).Very recent data [29] point, however, towards a σ(np → K+)/σ(pp → K+) ratio of 3-4. A furthercomplication is the production of the K+ in the pp → K+NΣ channel, which has been foundto be small [28], but this might be a consequence of the final state interaction. The differentσ(BB → K+) cross sections employed are presented in fig. 7 and Table 3. Top panels of

Program σ(NN → K+) σ(N∆ → K+) (σ∆∆ → K+)Barz [27] [27] 0Chen [27] [27] [27]Fuchs [30] (Λ) [27] Σ [27] [27]

Gaitanos [30] (Λ) [27] Σ [27] [27]HSD (Cassing) [12] [27] [27]

IQMD (Hartnack) [30] (Λ) [27] Σ [27] [27]Larionov [27] [27] [27]

Table 3. The K+ cross sections used in the programs.

Figure 7. The isospin averaged cross sections NN → K+.

fig. 8 and fig. 9 present, respectively, the K+ rapidity and transverse momentum distributionobtained for those cross sections. Also here the URQMD calculations (Reiter) cannot directly becompared with the others. In this approach, the K+ are created by two body resonance decays

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 9

0.00

0.02

0.04

0.06

0.08 Barz/Wolf

Cassing

Reiter

Hartnack

K+

Au+Au 0.96 AGeV

d

N/d

y

no potential

0.0

0.2

0.4

Au+Au 1.48 AGeV

0.0

0.1

0.2

0.3

Larionov

Chen

Fuchs

Gaitanos

Ni+Ni 1.93 AGeV

-2 -1 0 1 20.00

0.02

0.04

y-yc.m.

with potential

-2 -1 0 1 20.0

0.1

0.2

0.3

y-yc.m.

-2 -1 0 1 20.0

0.1

0.2

y-yc.m.

Figure 8. Final K+ rapidity distribution at b = 1 fm and with an enforced ∆ lifetime of 1/120MeV (top row without, bottom row with KN potential) in the different approaches.

whereas in all other approaches the K+ production is a process with three bodies in the final state:BB → K+Λ(Σ)B. We see that URQMD is not suited for 1 AGeV but already at 1.5 AGeV itdescribes the spectra in a reasonable way although spectra measured in elementary collisions pointtowards a three - body process. In order to understand the 50% differences of the predictionsone has to realize that the K+ are produced at subthreshold energies in the Au+Au collisions.Therefore the nucleons, which are involved in the K+ production, had already collisions before.This encodes in the production yield the whole history of the reaction. In addition, the K+

production cross section rises exponentially close to the threshold and therefore small differencesin the parametrization of the Fermi motion of the nucleons or in the potential of the hyperon changethe cross section considerably. Some programs contain, in addition, the channel πB → Y K+, butusually this channel does not contribute a lot at these energies due to its large threshold and thefew pions as compared to baryons.

Kaons change their properties in the medium. This is the prediction of many theoreticalapproaches which range from chiral perturbation theory to Nambu Jona-Lasinio modelcalculations [31]. All these theories should give reliable predictions if the density is low but sufferfrom the fact that the extrapolations to higher densities are less and less well founded. For ourpurpose, it is sufficient to parametrize the density dependence by

ωK(ρB , k) =√

m∗K(ρB)2 + k2, (5)

where the effective mass is

m∗K(ρB) = m0

K

(

1− αρBρ0

)

, (6)

with ρB and ρ0 being the baryon density and the normal nuclear matter density, respectively.Table 4 displays the different values of α employed in the calculations: It has been suggested in

ref. [32] to compare the ratio dσpC

dpt/ dσpA

dpt, which has been measured [33], with calculations using

different K+N potentials [33], to fix the K+ potential at normal nuclear matter density.

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 10

1E-4

1E-3

0.01

0.1

1 Au+Au 0.96 AGeV

K+

dN

/pTdp

T [1/G

eV

2]

no potential

1E-4

1E-3

0.01

0.1

1

10 Au+Au 1.48 AGeV

1E-4

1E-3

0.01

0.1

1

Barz/Wolf

Cassing

Reiter

Hartnack

Larionov

Chen

Fuchs

Gaitanos

Ni+Ni 1.93 MeV

0.0 0.2 0.4 0.6 0.8 1.0 1.21E-4

1E-3

0.01

0.1

pT [GeV/c]

with potential

0.0 0.2 0.4 0.6 0.8 1.0 1.21E-3

0.01

0.1

1

pT [GeV/c]

0.0 0.2 0.4 0.6 0.8 1.0 1.21E-4

1E-3

0.01

0.1

1

pT [GeV/c]

Figure 9. Final K+ transverse momentum distribution at b=1 fm, |ycm < 0.5| and with anenforced ∆ lifetime of 1/120 MeV (top row without, bottom row with KN potential) in thedifferent approaches.

Program α[MeV ] for K+ (K−)Barz -0.05 (0.16)Chen -0.04 (0.22)Fuchs -0.07

HSD (Cassing) -0.04 (0.10)IQMD (Hartnack) -0.075 (0.22)

Larionov -0.06 (0.2)

Table 4. Parameters for the density dependence of the in medium kaon masses m∗K

(eq.6).

Despite of the moderate increase of the ωK+(k = 0) with density, the potential changes theK+ yield considerably, as one sees in figs. 8 and 9. This is due to the fact that kaons are producedat high density when the mean-free-path is short and therefore the ∆’s have a higher chance tocollide with a nucleon. Depending on the density dependence of the potential, we see a decreaseof the total K+ yield of about a factor of 2 for the heavy systems and less for the light system ifthe potential is switched on. The form of the transverse momentum distribution by Fuchs differsfrom those from other calculations because he uses a covariant form of nuclear current instead ofonly its forth component, the baryon density, as the other approaches.

The elementary K+ creation process is azimuthally isotropic. It has been argued that theazimuthal distribution of the K+ contains information on the K+ nucleon potential [34]. This isbased on the fact that K+ emitted into the azimuthal angle φ = 90◦ come from highly compressedmatter whereas those being emitted near φ = 0◦(180◦) have traversed the cold spectator matter.Therefore one expects an enhancement around φ = 90◦ due to the more repulsive potential. Suchan effect may, however, be superimposed by the consequence of the different path lengths in matter.Fig.10 shows the result. The majority of the programs produce a different azimuthal distributionwithout (top row) and with (bottom row) K+N potential. Without the potential, the distribution

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 11

is more isotropic and the expected enhancement is seen if the repulsive potential is active.

0.6

0.8

1.0

1.2

K+

Au+Au 0.96 AGeV

dN

/dφ

[no

rma

lize

d]

no potential

Au+Au 1.48 AGeV

NI+Ni 1.93 AGeV

0 30 60 90 120 1500.6

0.8

1.0

1.2

with potential

0 30 60 90 120 150

φ0 30 60 90 120 150 180

Barz/Wolf

Cassing

Hartnack

Larionov

Chen

Figure 10. Final K+ azimuthal distribution at b=5 fm, |ycm < 0.5| and with an enforced∆ lifetime of 1/120 MeV (top row without, bottom row with KN potential) in the differentapproaches.

7. K− Mesons

The production process for the K− mesons in nuclear reactions is much more complex than thatfor the K+. The reason for this is their s quark content. On the one hand it is much harder toproduce a K− in elementary pp reactions because of its higher threshold (EKin = 2.513 GeV forpp → K+K−pp), and on the other hand a K− can exchange its s quark with strange baryons(K−N → Λ(Σ)π). The cross sections for these exothermic reactions are very large. The inversereaction Λ(Σ)π → K−N has been identified as the dominant source of the K− production inheavy ion reactions at the energies we consider here [35]. This process is absent in elementarypp interactions and therefore the K− yield in heavy ion collisions is orders of magnitudes largerthan that in pp collisions for energies not too far away from the threshold. This process couplesthe K− to the Λ(Σ) and hence to the K+ production [36, 37, 38] and explains therefore theexperimental observation [39] that the K+ to K− ratio is independent of the impact parameterThe K− production in baryon-baryon interactions suffer from the large threshold and takes placeonly early when the system is dense. Thereafter theK− gets easily reabsorbed. In figs. 11 and 12 wedisplay the K− rapidity and transverse momentum distributions. The URQMD calculations sufferfrom the fact that they produce too few strange baryons (produced in the BB → K+Y reactions) ascan been seen from the low K+ multiplicity. The K− channel confronts the simulation programswith the challenge to calculate the K−N and the Λπ cross sections at values of

√s which are in

free space below the threshold. These problem has been solved differently by the different groups.In view of the complexity of the production processes, the results of those codes, which

can be directly compared, are rather similar, especially at higher energies. At higher energies,the transverse momentum distributions are also rather similar. At lower energies the HSD codeproduces a somehow softer distribution. Also the K− changes its properties in the medium. As

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 12

0.0

0.5

1.0

1.5

2.0

K--

Au+Au 0.96 AGeV

d

N/d

y

no potential

x10-3

0.0

0.5

1.0

1.5

2.0 Barz/Wolf

Cassing

Reiter

Au+Au 1.48 AGeV

x10-2

0.0

0.5

1.0

1.5

Hartnack

Larionov

Chen

Ni+Ni 1.93 AGeV

x10-2

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

y-yc.m.

with potential

x10-3

-2 -1 0 1 20.0

0.5

1.0

1.5

y-yc.m.

x10-2

-2 -1 0 1 20.0

0.5

1.0

y-yc.m.

x10-2

Figure 11. Final K− rapidity distribution at b=1 fm and with an enforced ∆ lifetime of 1/120MeV (top row without, bottom row with KN potential) in the different approaches.

pointed in [40], due to the strange resonances this change is rather complex.In fact the detailedcalculations [5], based on the well constrained KN interaction from [41], raise the question whetherthe K− can still be treated as a quasi particle at high densities and temperatures. On the otherhand, most of the observed K− are produced not too far away from the surface and therefore themultiplicity should not be too much influenced by the kaon properties at high density. Currently,one has approached this problem by using the same form of the mean field potential as for the K+

with the parameters displayed in table 4. The energy ωK−(k = 0) decreases with increasing densityand therefore one expects an increase of the K− yield if the potential is switched on. This is indeedobserved in the bottom figures of the figs.11 and 12. The enhancement seen when the K− potentialis switched on is different in the different approaches. An important part of this difference comesfrom different extrapolations of the known K−N cross section, which diverges at small relativemomenta to the

√s <

√sthreshold region. A solid theoretical basis for this extrapolation, which is

necessary because the mass of the K− decreases and hence the threshold is lowered, is not at handpresently.

8. Lasting results

One of the most challenging motivations for studying heavy ion collisions has been the search forthe amount of energy which is needed to compress hadronic matter. This went under the slogan”Search of the nuclear equation of state”. Earlier ideas to use pions or the in-plane flow of baryonsas an experimental signal for the compressibility of matter, have not led to unambiguous results.The pions turned out to be not sensitive because most of them are produced when the systemhas already expanded to a low density. The in-plane flow is small at densities reachable in thesereactions and it depends on the pressure which is proportional to the density gradient. Nuclearsurface properties, like density gradients, are in general much more difficult to simulate than bulkproperties.

It has been proposed to use K+ for this purpose because at these subthreshold energies they

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 13

1E-5

1E-4

1E-3

0.01

Au+Au 0.96 AGeV

K--

dN

/pTdp

T [1/G

eV

2]

no potential

1E-5

1E-4

1E-3

0.01

0.1

Au+Au 1.48 AGeV

1E-5

1E-4

1E-3

0.01

0.1

Barz/Wolf

Cassing

Reiter

Hartnack

Larionov

Chen

Ni+Ni 1.93 MeV

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1E-5

1E-4

1E-3

0.01

pT [GeV/c]

with potential

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1E-5

1E-4

1E-3

0.01

0.1

pT [GeV/c]

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1E-5

1E-4

1E-3

0.01

0.1

pT [GeV/c]

Figure 12. Final K− transverse momentum distribution at b=1 fm, |ycm < 0.5| and withan enforced ∆ lifetime of 1/120 MeV (top row without, bottom row with KN potential) in thedifferent approaches.

test the nuclear environment prior to production [42]. To get rid of many of the uncertainties ithas been proposed by the KaoS collaboration [43] to use not the absolute yield but to compare aheavy with a light system where compression is virtually absent. Indeed, one has found that theexcitation function of this ratio is rather sensitive to the compressibility of the nuclear potentialand that the experimental results point strongly towards a nuclear potential with a compressibilityκ ≤ 280MeV . Such a potential has usually been dubbed as the ”soft equation of state”. In themeantime, one has tested whether this result is robust, i.e. whether it remains valid if one variesthe unknown or little known input such as the ∆ lifetime or the ∆N → K+ cross section with theconclusion that only a strongly density dependent K+ production cross section may invalidate thisresult. Such a cross section, however, has never been proposed in elementary calculations. To whichdegree this result is stable against a change of the ∆N → K+ cross section is shown in the left panelof fig. 13 where the different cross sections vary by a factor of 2. The results appear fairly robust.One should stress, though, that we find here that nuclear matter is relatively easy to compress.The compressibility is compatible with the results obtained by analyzing giant resonances, i.e. thecompression of nuclei at very low temperature and around the equilibrium density. However, thereactions discussed here yield high densities and temperatures and the system contains mesons andresonances. Thus we test here a completely different region in the T-ρ plane.

9. Conclusion

Multiplicity, rapidity and transverse momentum distributions from different programs simulatingheavy ion reactions at kinetic energies of around 1 AGeV have been compared. If the same inputsare used, the programs yield quite similar results but differ in details. Some of the differencesare understood in terms of the implementation of different physical approaches in the programs,and others, less important, remain to be explained. Despite of these differences the programsagree on some important physical results: The pions are almost exclusively produced by resonance

Transport Theories for Heavy Ion Collisions in the 1 AGeV Regime 14

1

Figure 13. Normalized ratio of the K+ yield in Au+Au to C+C reactions as a function thebeam energy for two different equations of states. Filled circles with errors represent the data ofthe KaoS collaboration [43]. Other symbols with lines represent calculations of ref. [21] on theleft panel and of ref. [19] on the right panel for either the ∆N → K+ cross section of ref.[27] orfor (σ(∆N → K+) = 3/4 σ(NN → K+)) employed earlier.

decay. Subthreshold meson production is due to the same reaction as in free space and becomespossible because at least one of the collision partners has gained energy in collisions before. Theyprograms furthermore agree on the stopping of matter, the dependence of the K+ squeeze outon its in-medium potential and on the softness of the density dependence of the effective nuclearpotential.

Acknowledgment We would like to thank Dr. H. Oeschler for many valuable discussions andthe ECT ∗ in Trento, Italy, for hosting a workshop which was the starting point of this comparison.

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